Solve the equation: $\;$ $\log_4 \left(x + 3\right) - \log_4 \left(x - 1\right) = 2 - \log_4 8$
Given equation: $\;\;$ $\log_4 \left(x + 3\right) - \log_4 \left(x - 1\right) = 2 - \log_4 8$
i.e. $\;$ $\log_4 \left(\dfrac{x + 3}{x - 1}\right) + \log_4 8 = 2$
i.e. $\;$ $\log_4 \left[\dfrac{8 \left(x + 3\right)}{x - 1}\right] = 2$
i.e. $\;$ $\dfrac{8 \left(x + 3\right)}{x - 1} = 4^2 = 16$
i.e. $\;$ $8x + 24 = 16x - 16$
i.e. $\;$ $8x = 40$ $\implies$ $x = 5$
$\therefore \;$ The solution to the given equation is $\;\;$ $x = \left\{5 \right\}$